is combined with a pressure force based on the ideal gas law. Finite element simulation This is a more physically accurate approach, which uses the widely used finite element method to solve the partial differential equations which govern the dynamics of an elastic material. The body is modeled as a three-dimensional elastic continuum by breaking it into a large number of solid elements which fit together, and solving for the stresses and strains in each element using a model of the material. The elements are typically tetrahedral, the nodes being the vertices of the tetrahedra (relatively simple methods exist to tetrahedralize a three dimensional region bounded by a polygon mesh into tetrahedra, similarly to how a two-dimensional polygon may be triangulated into triangles). The strain (which measures the local deformation of the points of the material from their rest state) is quantified by the strain tensor . The stress (which measures the local forces per-unit area in all directions acting on the material) is quantified by the Cauchy stress tensor . Given the current local strain, the local stress can be computed via the generalized form of Hooke's law: where is the elasticity tensor, which encodes the material properties (parametrized in linear elasticity for an isotropic material by the Poisson ratio and Young's modulus). The equation of motion of the element nodes is obtained by integrating the stress field over each element and relating this, via Newton's second law, to the node accelerations. Pixelux (developers of the Digital Molecular Matter system) use a finite-element-based approach for their soft bodies, using a tetrahedral mesh and converting the stress tensor directly into node forces. Rendering is done via a form of free-form deformation. Energy minimization methods This approach is motivated by variational principles and the physics of surfaces, which dictate that a constrained surface will assume the shape which minimizes the total energy of deformation (analogous to a soap bubble). Expressing the energy of a surface in terms of its local deformation (the energy is due to a combination of stretching and bending), the local force on the surface is given by differentiating the energy with respect to position, yielding an equation of motion which can be solved in the standard ways. Shape matching In this scheme, penalty forces or constraints are applied to the model to drive it towards its original shape (i.e. the material behaves as if it has shape